The sharp Faber-Krahn inequality

(ENS Lyon)

Abstract. The classical Faber-Krahn inequality asserts that balls (uniquely) minimize the first eigenvalue of the Dirichlet-Laplacian among sets with given volume. I will show a sharp quantitative enhancement of this result, confirming a conjecture by Nadirashvili and Bhattacharya-Weitsman: \[ \lambda_1(\Omega)-\lambda_1(B_1)\ge c_N \mathcal A (\Omega)^2\qquad \text{for all \(\Omega\subset \mathbb R^N\) such that \(|\Omega|=|B_1|\)}, \] where \(\mathcal A(\Omega)\) is the Frankel asymmetry of a set: \[ \mathcal A(\Omega)=\inf_{x_0\in \mathbb R^N} |\Omega \Delta B_1(x_0)|. \]

More generally, the result applies to every optimal Poincar\'e-Sobolev constant for the embeddings \(W^{1,2}_0(\Omega)\hookrightarrow L^q(\Omega)\). (Joint work with L. Brasco and B. Velichkov).